On the nonlinear reflection of a gravity wave at a critical level. Part 3

Abstract

In part 2 (Brown & Stewartson 1982) of this paper we set out the linearized solution of the critical layer that is expected to hold some time after the forced internal gravity wave of small amplitude e is initiated at an infinite distance above the shear layer. This differed from that presented in part 1 (Brown & Stewartson 1980) in that in part 2 we exploited to a much greater extent the fact that the Richardson number J was large and obtained the solution in a form consisting of explicit functions rather than infinite integrals. Also it was demonstrated that at times t = O(1) the wave did not penetrate beyond a certain level in the critical layer, and that critical-layer noise only was created above this line and transmitted below it. In this paper we examine the development of this solution on a longer time scale $\tau (\propto \epsilon^{\frac{2}{3}}t)$ and show how the reflection and transmission coefficients which are exponentially small, \[ O\{\exp(-(J-\frac{1}{4})^{\frac{1}{2}}\pi)\}, \] when τ = 0 increase with time. As in part 1 we obtain a reflection coefficient for the first harmonic that is O(τ³), and because of the simpler formulation of the linearized solution are able to obtain a reflected second harmonic. These harmonics appear as complementary functions that are induced by singularities in the particular integrals of the equations. It is shown that the interaction between the initial-noise term in the lower part of the critical layer and an induced-noise term at the sixth stage of the expansion will eventually lead to a transmitted wave. This appears at the ninth stage of the expansion and its transmission coefficient is O(τ¹²) though it is not explicitly calculated here.